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Properties

Label 2-396-12.11-c1-0-15
Degree $2$
Conductor $396$
Sign $-0.887 + 0.460i$
Analytic cond. $3.16207$
Root an. cond. $1.77822$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.929 − 1.06i)2-s + (−0.272 + 1.98i)4-s − 3.24i·5-s − 1.06i·7-s + (2.36 − 1.55i)8-s + (−3.45 + 3.01i)10-s + 11-s − 0.185·13-s + (−1.13 + 0.985i)14-s + (−3.85 − 1.07i)16-s − 1.21i·17-s − 0.402i·19-s + (6.42 + 0.884i)20-s + (−0.929 − 1.06i)22-s − 8.43·23-s + ⋯
L(s)  = 1  + (−0.657 − 0.753i)2-s + (−0.136 + 0.990i)4-s − 1.45i·5-s − 0.400i·7-s + (0.836 − 0.548i)8-s + (−1.09 + 0.953i)10-s + 0.301·11-s − 0.0514·13-s + (−0.302 + 0.263i)14-s + (−0.962 − 0.269i)16-s − 0.294i·17-s − 0.0923i·19-s + (1.43 + 0.197i)20-s + (−0.198 − 0.227i)22-s − 1.75·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $-0.887 + 0.460i$
Analytic conductor: \(3.16207\)
Root analytic conductor: \(1.77822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :1/2),\ -0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189685 - 0.777082i\)
\(L(\frac12)\) \(\approx\) \(0.189685 - 0.777082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.929 + 1.06i)T \)
3 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 3.24iT - 5T^{2} \)
7 \( 1 + 1.06iT - 7T^{2} \)
13 \( 1 + 0.185T + 13T^{2} \)
17 \( 1 + 1.21iT - 17T^{2} \)
19 \( 1 + 0.402iT - 19T^{2} \)
23 \( 1 + 8.43T + 23T^{2} \)
29 \( 1 + 7.33iT - 29T^{2} \)
31 \( 1 + 8.06iT - 31T^{2} \)
37 \( 1 + 0.871T + 37T^{2} \)
41 \( 1 + 6.18iT - 41T^{2} \)
43 \( 1 - 9.72iT - 43T^{2} \)
47 \( 1 + 4.14T + 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 - 4.64T + 61T^{2} \)
67 \( 1 + 2.66iT - 67T^{2} \)
71 \( 1 - 9.80T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 - 15.1iT - 89T^{2} \)
97 \( 1 - 2.18T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.90759973936180673486589523473, −9.750228927028808570937841647920, −9.309704985066995631479791163325, −8.216866151254820678598865274057, −7.66209248785769377492378944997, −6.09857040028372302510689411482, −4.64264298609389426526313846924, −3.88839786124687670639656945679, −2.09270257683690935699028806347, −0.65529544868567777771118110202, 2.03478821721680913253810186897, 3.57133310778741382314957027886, 5.21703740929479802338136022387, 6.33758362517378647095157571313, 6.88876810547278730511190813485, 7.913479830766678379650913108792, 8.823081027159327241873810506558, 9.942850435695832112522480957535, 10.52262979014230277668716904932, 11.36708926723157417799422536858

Graph of the $Z$-function along the critical line